nLab chromatic localization




The Bousfield localization of spectra L E(n)L_{E(n)} at nnth Morava E-theory is called chromatic localization. The tower of chromatic localizations as nn ranges is called the chromatic tower, leading to the chromatic filtration. This is the subject of chromatic homotopy theory.


Relation to formal groups

Chromatic localization on complex oriented cohomology theories is like the restriction to the closed substack

FG n+1 FG×Spec (p) \mathcal{M}_{FG}^{\leq n+1} \hookrightarrow \mathcal{M}_{FG} \times Spec \mathbb{Z}_{(p)}

of the moduli stack of formal groups on those of height n+1\geq n+1.

(e.g. Lurie, lect 22, above theorem 1)

In this way the localization tower at the Morava E-theories exhibits the chromatic filtration in chromatic homotopy theory.


chromatic homotopy theory

chromatic levelcomplex oriented cohomology theoryE-∞ ring/A-∞ ringreal oriented cohomology theory
0ordinary cohomologyEilenberg-MacLane spectrum HH \mathbb{Z}HZR-theory
0th Morava K-theoryK(0)K(0)
1complex K-theorycomplex K-theory spectrum KUKUKR-theory
first Morava K-theoryK(1)K(1)
first Morava E-theoryE(1)E(1)
2elliptic cohomologyelliptic spectrum Ell EEll_E
second Morava K-theoryK(2)K(2)
second Morava E-theoryE(2)E(2)
algebraic K-theory of KUK(KU)K(KU)
3 …10K3 cohomologyK3 spectrum
nnnnth Morava K-theoryK(n)K(n)
nnth Morava E-theoryE(n)E(n)BPR-theory
n+1n+1algebraic K-theory applied to chrom. level nnK(E n)K(E_n) (red-shift conjecture)
\inftycomplex cobordism cohomologyMUMR-theory


  • Mark Hovey, Hal Sadofsky, Invertible spectra in the E(n)E(n)-local stable homotopy category (pdf)

Last revised on April 7, 2014 at 01:29:49. See the history of this page for a list of all contributions to it.